A Simplified Steady-State Model for Predicting the Energy Consump

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International Refrigeration and Air ConditioningConference

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2008

A Simplied Steady-State Model for Predicting theEnergy Consumption of Household Refrigerators

and FreezersJoaquim M. GonzalvesFederal University of Santa Catarina

Christian J. L. HermesFederal University of Santa Catarina

Claudio MeloFederal University of Santa Catarina

Fernando T. Knabben

Federal University of Santa Catarina

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Gonzalves, Joaquim M.; Hermes, Christian J. L.; Melo, Claudio; and Knabben, Fernando T., "A Simplied Steady-State Model forPredicting the Energy Consumption of Household Refrigerators and Freezers" (2008).International Reigeration and Air ConditioningConference. Paper 944.hp://docs.lib.purdue.edu/iracc/944
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International Refrigeration and Air Conditioning Conference at Purdue, July 14-17, 2008

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Joaquim M. GONALVES*, Christian J. L. HERMES, Cludio MELO, Fernando T. KNABBEN

POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics,

Federal University of Santa Catarina, 88040-900, Florianpolis, SC, Brazil

Phone: +55 48 3234 5691, e-mail: [email protected]

(*)HVAC&R Research Group, Federal Center of Technological Education of Santa Catarina,

Rua Jos Lino Kretzer 608, 88103-310, So Jos, SC, Brazil

ABSTRACT

A simplified model to assess the energy performance of vapor compression refrigerators and freezers is presentedherein. The model consists of first-principles algebraic equations adjusted with experimental information obtained

from the refrigeration system under study. The experimental work consisted of controlling and measuring the system

and component operating conditions in order to gather key information for the development and validation of the

model. The methodology showed similar accuracy to that using more sophisticated dynamic simulation codes, but

with lower computational costs. When compared to experimental data, the model predicted AHAM energy

consumption tests within a 5% deviation band. A sensitivity analysis considering the number of tube rows in the

condenser coil, the number of fins in the evaporator coil and the compressor stroke is also reported. The

refrigeration system under study was a top-mount Combi 600-liter refrigerator, on-off controlled by the fresh-food

compartment temperature.

1. INTRODUCTION

Component matching and energy assessment of household refrigeration systems are usually carried out throughcostly, time consuming standardized test procedures (e.g., AHAM HRF-1, 2004; ISO15502, 2005). In order to speed

up the product development process, first-principles simulation models have been employed to simulate the thermo-

hydrodynamic behavior of these appliances. Steady-state and transient approaches have both been used, the former

for component matching (e.g., Davis and Scott, 1976; Abramson et al., 1990; Klein et al., 1999; Gonalves and

Melo, 2004), and the latter essentially to assess energy performance (e.g., Melo et al. 1988, Jansen et al. 1988,

Jakobsen, 1995; Hermes and Melo, 2006).

In a previous study, Gonalves and Melo (2004) investigated the steady-state behavior of a top-mount refrigerator

by measuring and modeling the performance characteristics of each of its components. The models were based on

the mass, energy and momentum conservation principles and also on empirical data. Measurements of the relevant

variables were taken at several positions along the refrigeration loop, generating performance data not only for the

whole unit but also for each of the cycle components. The experiments were planned and performed following a

statistical methodology and considering 13 independent variables that led to over 160 data runs. The modelpredictions for the cooling capacity and power consumption, when compared with experimental data, were within a

10% deviation band. The model was also used to simulate the effect of the system parameters on the refrigerator

performance in an attempt to minimize the power consumption for a given internal air temperature. However, the

steady-state model of Gonalves and Melo (2004) is not suitable for predicting the refrigerator energy consumption,

since this is dependent not only on the power consumption, but also on the compressor runtime.

Later, Hermes and Melo (2006) put forward a first-principles model for simulating the transient behavior of

household refrigerators. The model was used to simulate a typical top-mount refrigerator, where the compressor is

on-off controlled by the freezer temperature, while a thermo-mechanical damper is used to set the fresh-food


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compartment temperature. Innovative modeling approaches were proposed for each of the refrigerator components

(condenser, evaporator, compressor, capillary tube-suction line heat exchanger, and refrigerated compartments).

Numerical predictions were compared to experimental data showing a reasonable level of agreement for the whole

range of operating conditions, including the start-up and cycling regimes. The system energy consumption was

found to be within 10% agreement with the experimental data, while the compartment air temperatures were

predicted with a maximum deviation of 1C. The model was also used to assess the impact of some operational and

geometric parameters on the overall energy consumption. The code, however, requires a considerable amount ofCPU time, not being viable for optimization tasks that may require thousands of runs in a row.

In order to address this drawback, a simplified methodology for predicting the energy consumption of household

refrigerators and freezers with a similar accuracy, but with a substantially lower computational effort than that

required by the dynamic simulation code of Hermes and Melo (2006), is proposed herein. The model followed the

steady-state semi-empirical approach introduced by Gonalves and Melo (2004), but was adapted to estimate the

refrigerator runtime as a function of the cabinet thermal loads and cooling capacity.

2. MATHEMATICAL MODEL

The mathematical formulation follows closely that introduced by Gonalves and Melo (2004), according to which

the refrigeration system was divided into the following sub-domains: compressor, heat exchangers (evaporator and

condenser), capillary tube-suction line heat exchanger, evaporator and condenser fans, and refrigerated

compartments, as depicted in Fig. 1.

Figure 1. Schematic representation of a top-mount refrigerator

2.1 CompressorThe refrigerant enthalpy at the compressor discharge, h2, was calculated from an energy balance in the compressor

shell, as follows

( ) mQWhh kk+=

12

(1)

The compressor mass flow rate, m, was obtained from equation (2), where v stands for the volumetric efficiency,

1vNVm kv= (2)

The compression power, Wk, was obtained from equation (3), where g corresponds to the compressor overall

efficiency,

( )gs,k hhmW 12 = (3)

Qc

Qk

Wk

Qe


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The heat released by the compressor shell to the surroundings, Qk, was calculated from

( )akk TTUAQ = 2 (4)

The volumetric and overall efficiencies, v and g, and the overall thermal conductance, UAk, were all obtained fromexperimental tests carried out with the refrigerator. The volumetric and overall efficiencies were fitted as linear

functions of the pressure ratio,pc/pe, while the thermal conductance UAk was assumed to be constant.

2.2 Capillary tube-suction line heat exchangerThe refrigerant enthalpy at the capillary tube exit, h4, was obtained from an energy balance in the capillary tube-

suction line heat exchanger (see Fig. 1),

1534 hhhh += (5)

The suction line exit temperature, T1, was calculated from equation (6), where the heat exchanger effectiveness, ,was derived from experimental tests carried out with the refrigerator,

( )5351 TTTT += (6)

2.3 Heat exchangers

2.3.1 Thermal model

The heat exchangers (condenser and evaporator) were divided into sub-regions according to the thermodynamic

state of the refrigerant. The condenser was divided into three different regions (superheating, saturation and

subcooling), whilst the evaporator was divided into two regions (saturation and superheating). The heat exchanger

coils were assumed to be straight and horizontal, with uniformly distributed fins. The coils were divided into N

control volumes. The heat transferred was calculated by applying equations (7) to (9) to each of the control volumes,

( ) maxminx,ox,ix TChhmQ == (7)

where is the temperature effectiveness of the cross-flow heat exchanger, expressed as a function of the number oftransfer units, NTU=UAx/Cmin, and of the ratio between the refrigerant and air-side thermal capacities,

Cratio=Cmin/Cmax (Kays and London, 1998). The effectiveness of the two and single-phase flow regions were

calculated as follows,

( )NTUexp = 1 (8)

( )[ ]{ }ratioratio CNTUCexpexp = 11 (9)

The overall thermal conductance, UAx, was calculated from

( ) ( ) 111 += eeeiix AAUA (10)

The tube wall and the tube-fin contact thermal resistances were both neglected. An infinite internal heat transfer

coefficient, i, was adopted for the two-phase flow regions, whilst Gnielinskis correlation (1976) was employed for

the single-phase regions. The length of each flow region was determined by comparing the local refrigerant enthalpy

with that of saturated vapor or saturated liquid. The wire-and-tube condenser and the tube-fin no-frost evaporator

external heat transfer coefficients, e, were calculated using the correlations of Petroski and Clausing (1999) andKim et al. (1999), respectively. The fin efficiencies were determined by Schmidts method (1945).

2.3.2 Air flow model

The air-side pressure drop of the heat exchangers were calculated from

2

xaxx VKp= (11)

The bypass air was also taken into account by considering that

2

xabypassbypassx VKpp == (12)


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The overall air flow rate Vx was calculated iteratively from equations (11) and (12). The condenser pressure loss

coefficient Kx was obtained from tests carried out in a wind-tunnel facility, whereas the evaporator pressure drop

was estimated by Kim et al.s correlation (1999). The condenser and evaporator fans were modeled by 6th

-order

polynomial fits ofpxfand xf as functions of the air flow rate, Vx. The pumping power was calculated as follows,

xfxxfxf VpW = (13)

2.4 Refrigerated compartments

2.4.1 Thermal model

Figures 2 and 3 respectively show schematic representations of the energy and fluid flows within the refrigerated

compartments. The evaporator air mass flow rate, mef, is split into two air streams by a damper action; part of the air

is supplied to the freezer compartment, mfz, and part to the fresh-food compartment, mff. An energy balance

involving the evaporator, the freezer and fresh-food compartments yields (see Fig. 2),

( ) ( ) ( )fzffmfzafzefe TTRTTUAWQr +=1 (14)

( )( ) ( ) ( )fzffmffaffefe TTRTTUAWQr =1

1 (15)

where r=mfz/mef is the freezer air flow rate fraction, UAfz and UAff are the overall thermal conductances of the freezer

and fresh-food compartments, andRm is the mullion thermal resistance, defined as

( ) mpaefm UAcmrrR += 11 (16)

Figure 2. Heat flow within the refrigerated

compartments

Figure 3. Fluid flow within the refrigerated

compartments

It is worth noting that the compartment temperatures are design conditions (ISO: Tff=5C and Tfz=-18C; AHAM

Tff

=7.2C and Tfz

=-15C) that need to be specified by the user. Equations (14) to (16) must then be solved for the

freezer air fraction, r, a parameter that balances the air flow between the freezer and fresh-food compartments.

Finally, the air temperature at the evaporator inlet is calculated from

( ) fffzm TrrTT += 1 (17)

2.4.2 Air flow modelThe air-side pressure drop in each branch of the refrigerated compartments (see Fig. 3) was computed using

equation (11), where the pressure loss factor, Kx, was obtained from wind-tunnel experiments. The pressure loss

from the evaporator inlet to the fan outlet is given by


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655414 = pppp ef (18)

where pef is the fan pressure rise, and p4-1 corresponds to the pressure drops in both the freezer and fresh-food

compartments,

43322114 ++= pppp (19)

The hydrodynamic coupling between the evaporator, the evaporator fan, and the refrigerated compartments is givenby equations (18) and (19) which, when solved, provide the evaporator air flow rate.

2.5 Energy consumption calculationThe energy consumption of a thermostat-controlled refrigerator is calculated by integrating the overall power

consumption during a running cycle. Assuming that both the thermal load, Qt, and the cooling capacity, Qe, are

nearly constant during the cycling regime, the energy consumption can be estimated by an approximated runtime

ratio, calculated from the following energy balance over a running cycle,

( )on on off t et t t Q Q + (20)

Thus, the overall energy consumption can be easily calculated from

cfefk

WWWEnergy ++=(21)

where Wk, Wefand Wcfare the power required by the compressor, evaporator fan and condenser fan, respectively.

2.6 Determination of the working pressuresTwo additional equations are required to determine the evaporating and condensing pressures. In general, the

working pressures are obtained implicitly and iteratively from the following mass balances,

0'= mm (22)

0=n

nMM

(23)

where m and mare the compressor and the expansion device (not modeled) mass flow rates, respectively, Mis theactual refrigerant charge and Mn is the calculated amount of refrigerant in each of the n components of the

refrigeration loop (not modeled). It is worth noting that equations (22) and (23) are strongly non-linear functions ofthe working pressures, leading to time-consuming solutions and also to convergence issues. In order to keep a

reasonable level of complexity, equations (22) and (23) were replaced by fixed values of refrigerant superheating

and subcooling at the evaporator and condenser exits, respectively. The working pressures were then calculated

straightforwardly from,

supsate TTpp = 5 (24)

( )subsatc TTpp += 3 (25)

This procedure not only eliminates potential convergence issues, but also brings the numerical analysis closer to the

design practice, where both the capillary tube and the refrigerant charge are adjusted a posteriori to guarantee a

certain degree of superheating and subcooling.

3. SOLUTION ALGORITHM

The solution algorithm consists of two iterative loops. In the outer loop, the condensing and evaporating pressures,

the refrigerant temperature at the compressor inlet and the air temperature at the evaporator inlet are calculated by

the Newton-Raphson method. In the inner loop, a successive substitution strategy was adopted for each of the

system components. Thus, for a given set of values for pe,pc, h1 and Tm, the compressor sub-model calculates h2, the

condenser sub-model estimates h3 and T3=T(pc,h3), the internal heat exchanger sub-model calculates h4 and T1, and

the evaporator sub-model calculates h5 and T5=T(pe,h5). Finally, the cabinet thermal and hydraulic models are solved

to estimate both r and . The calculation procedure continues until convergence is achieved. The code wasimplemented using the EES platform (Klein, 2002) connected to REFPROP7 software (Lemmon et al., 2002).


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4. EXPERIMENTAL WORK

The refrigerator was instrumented and installed inside an environmental chamber with controlled air temperature

and humidity. Type T thermocouple probes immersed in the refrigerant flow passage and absolute pressure

transducers were installed at three points along the refrigeration loop: compressor suction and discharge ports, and

the condenser exit section. A Coriolis-type mass flow meter was installed in the discharge line. The outside air

temperature was measured by 5 thermocouples placed around the refrigerator. The freezer and the fresh foodcompartment air temperatures were measured by 3 and 5 thermocouples placed inside the compartments,

respectively. Air-side temperatures at the heat exchanger inlet and outlet ports were also measured. The compressor,

evaporator fan and condenser fan power consumptions were also monitored. Tests were carried out before and after

the instrumentation set-up to check for any effect on system performance. The system employed HFC-134a as the

working fluid (110 g).

Firstly, steady-state tests were carried out for three different ambient temperatures 25C, 32C and 43C to acquire

data required to adjust the model empirical parameters (e.g., cabinet thermal conductance). Later, 10 energy

consumption tests (ANSI/AHAM, 2004) were carried out with different thermostat and damper settings to validate

the model. In addition, the air-side pressure drops at several parts of the refrigerated compartments and also along

the heat exchangers (condenser and evaporator) were evaluated using a wind-tunnel test facility (Barbosa et al.,

2008). The wind-tunnel was also employed to evaluate the characteristic curves of both fans.

5. RESULTS

The model was validated by supplying the measured average cabinet air temperatures and the superheating and

subcooling degrees as input data. Table 1 compares the energy consumption model predictions with their

experimental counterparts. It can be noted that a consistent agreement was achieved, with all values falling inside a

5% error band. Some simulations were also performed to assess the impact of some design parameters on the

refrigerator performance. All simulations were carried out based on the AHAM test conditions, and using a fixed

subcooling and superheating of 3C and an internal heat exchanger effectiveness of 0.85.

Table 1. Predicted versus measured energy consumption values

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Figure 4 shows the energy consumption as a function of compressor stroke, but keeping the compressor efficiency

constant. It is worth noting that the resultant behavior is quasi linear and that the energy consumption dropped by

13% when the piston displacement was reduced from 5.96 to 3.77 cm3. This is so because the original compressor

cooling capacity is a bit excessive. The same kind of analysis was carried out, but using different compressors with

different capacities and efficiencies (see Fig. 5). The nominal capacities shown in Fig. 5 refer to the followingcompressor calorimeter testing conditions: Tc=55C, Te=-25C, T1=32C, and Ta=32C. Figure 5 shows that the

energy consumption dropped by 4.5%, when the original compressor (#2) was replaced by a lower capacity, same

efficiency compressor (#3). On the other hand, a lower capacity, lower efficiency compressor (#5) provided an

energy performance similar to that obtained with compressor #3. Moreover, a higher capacity, lower efficiency

compressor (#1) increased the energy consumption by 20%. Figure 6 explores the effect of the number of tube rows

in the condenser coil on the system performance. It can be seen that the energy consumption was reduced by 4%

when the number of condenser tube rows was increased from 18 to 24. It is clear that the augmentation of the heat

transfer surface overcame the increase in the air-side pressure drop, improving the condenser performance. Figure 7


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shows that the energy consumption practically does not change when more fins are added to the evaporator, but

increases by 2% when 10 fins are removed from the evaporator coil.

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.530

32

34

36

38

40

Compressor stroke [cm3]

Energyconsumption[kWh/month]

Original

-4.6%

-13%

34

36

38

40

42

44

Original

-4.5%

+20%


1 2 3 4 5

Compressor model

(2) Qe=185W, COP=1.63

(1) Qe=205W, COP=1.38

(3) Qe=155W, COP=1.65

(4) Qe=165W, COP=1.48

(5) Qe=120W, COP=1.48

Figure 4. Effect of compressor stroke on the energy

consumption: constant efficiency

Figure 5. Effect of compressor stroke on the energy

consumption: real compressors

9 12 15 18 21 24 2734

35

36

37

38

39

40

Number of tube rows


Original

-3.9%

+8.4%

0 20 40 60 80 10035

36

37

38

39

40

Number of evaporator fins


Original

Figure 6. Effect of condenser tube rows on the energy

consumption

Figure 7. Effect of evaporator fin on the energy

consumption

6. CONCLUSIONS

A simplified methodology for predicting the energy consumption of refrigerators and freezers using a first-principles

steady-state simulation model was proposed and validated against experimental AHAM energy consumption data.

The model predictions were compared with their experimental counterparts, showing agreement within a 5% error

band. Additionally, the model was used to assess the energy performance of a Combi refrigerator in terms of the

compressor stroke, the number of condenser tube rows, and the number of evaporator fins. It was shown that the

product energy consumption can be decreased by as much as 7.5% by using a lower capacity compressor and at the

same time adding 6 more tube rows in the condenser coil. Clearly, the economic feasibility of these modificationsneeds to be addressed. It should be emphasized that the model requires less than 1 min of CPU time (Intel Core 2

Duo 1.86GHz 2Gb RAM PC) to carry out an energy consumption prediction, a value substantially lower than that

required by more sophisticated dynamic simulation models.

NOMENCLATURE

RomanA area (m2)

C thermal capacity (W/K)

h specific enthalpy (J/kg)m mass flow rate (kg/s)

M refrigerant charge (g)


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N compressor speed (s-1)NTU number of transfer units (-)p pressure (Pa)Q heat transfer rate (W)

r fraction of air flow to the freezer compartment (-)T temperature (K)t time (s)

UA overall thermal conductance (W/K)v specific volume (m3/kg)V air flow rate (m3/s)Vk compressor displacement (m

3)W power (W)

Greek

heat transfer coefficient (W/m2K)

heat exchanger effectiveness (-)

efficiency (-)

density (kg/m3)

runtime ratio (-)

Subscripts

a air, ambientc condensing

e evaporating, externalff fresh-food compartment

fz freezer compartmenti internal, inletk compressor

m mixture, mullion

o outlet

x heat exchangers

xf heat exchanger fan

REFERENCES

Abramson DS, Turiel I, Heydari A, 1990, Analysis of Refrigerator-Freezer Design and Energy Efficiency by

Computer Modeling: DOE Perspective, ASHRAE Transactions, Vol. 96, Part I, pp.1354-1358

ANSI/AHAM HRF-1, 2004, Energy performance and capacity of household refrigerators, refrigerator-freezers andfreezers, American National Standards Institute, Washington-DC, USA

Barbosa Jr. JR, Melo C, Hermes CJL, 2008, A study of air-side heat transfer and pressure drop characteristics of

tube-fin no-frost evaporators, International Refrigeration and Air Conditioning Conference at Purdue, West

Lafayette-IN, USA, Paper 2310

Davis GL, Scott TC, 1976, Component Modeling Requirements for Refrigeration System Simulation: Large Effort,

Little Effect?, Compressor Technology Conference at Purdue, West Lafayette-IN, USA, pp.401-408

Gnielinski V, 1976, New equations for heat and mass transfer in turbulent pipe and channel flow, International

Chemical Engineering, Vol. 16, No. 2, pp. 359-368

Gonalves JM, Melo C, 2004, Experimental and numerical steady-state analysis of a top-mount refrigerator, Int.

Refrigeration Conference at Purdue, West Lafayette-IN, USA, Paper R078

Hermes CJL, Melo C, 2006, A dynamic simulation model for fan-and-damper controlled refrigerators, 11th

International Refrigeration Conference at Purdue, West Lafayette, USA, Paper R036

ISO/FDIS 15502, 2005, Household refrigerating appliances - Characteristics and test methods, InternationalOrganization for Standardization, Geneva, Switzerland

Jakobsen A, 1995, Energy optimisation of refrigeration systems: the domestic refrigerator a case study, PhD

thesis, Technical University of Denmark, Lyngby, Denmark

Jansen MJP, Kuijpers LJM, de Wit JA, 1988, Theoretical and experimental investigation of a dynamic model for

small refrigerating systems,IIR/IIF Meeting at Purdue, pp. 245-255

Kays WM, London AL, 1998, Compact heat exchanger, Krieger Publishing, Boca Raton-FL, USA

Kim NH, Youn B, Webb RL, 1999, Air-side heat transfer and friction correlations for plain fin-and-tube heat

exchangers with staggered tube arrangements, Transactions of the ASME, J. Heat Transfer, 121, pp.662-667

Klein FH, Melo C, Marco ME, 1999, Steady-State Simulation of an All Refrigerator, 20th International Congress of

Refrigeration, Sydney, 1999, Vol. III, Paper 073

Klein SA, 2002, EES Engineering Equation Solver Users Manual, F-Chart Software, Middleton, WI, USA

Lemmon EW, McLinden MO, Huber M.L., 2002, NIST Reference fluids thermodynamic and transport properties

REFPROP 7.0, National Institute of Standards and Technology, Gaithersburg, MD, USA

Melo C, Ferreira RTS, Negro COR, Pereira RH, 1988, Dynamic behaviour of a vapour compression refrigerator: a

theoretical and experimental analysis,IIR/IIF Meeting at Purdue, West Lafayette, USA, pp.98-106

Petroski SJ, Clausing AM, 1999, An investigation of the performance of confined, saw-tooth shaped wire-on-tube

condensers, Technical Report ACRC TR-153, University of Illinois, Urbana-IL, USA

Schmidt TE, 1945, La production calorifique ds surfaces munies dailettes, Bulletin de IIF, Annexe G-5

ACKNOWLEDGEMENTS

They authors duly acknowledge Whirlpool S.A., and the agencies CNPq and FINEP for their financial support.

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